Modelling strike duration distribution: a controlled Wiener process approach (Q2756666)

The author presents a method for derivation of the strike duration density distribution from a controlled Wiener process. Let the state of the model \(x(t)\) be some measure of disagreement between the strike parties. At moment \(t=0\) when the strike starts \(x(0) =x >d >0\), while \(d\) is the agreement or settlement point, usually the wage level or rate which settles the disputes. The author supposes that \(x(t)\) is defined by the stochastic differential equation \(dx(t) = u(t) dt +dw(t)\), where \(dw(t)\) is a Gaussian white noise with zero mean and variance \(\sigma^2.\) One wants to end the strike as soon as possible. This can be achieved by some control measures \(u(t)\) exercised by state authorities or negotiations between the parties.NEWLINENEWLINENEWLINEOne way to achieve this is to minimize the expected value of the cost function NEWLINE\[NEWLINEJ(x) = \int_0^T (qu^2/2) dt + KT(x),NEWLINE\]NEWLINE where \(q\) is the control cost and \(k\) is the terminal cost of the strike \(T(x)\) defined as \(T(x)=\inf\{ t\colon x(t) =d |x(0) =x>d\}.\) The author shows that under this assumptions the first exit time \(T_c\) of the controlled process has following density for \(t_c \geq 0\): NEWLINE\[NEWLINEf(x, t_c) =(x-d)(\sigma^2 2 \pi t_c^3)^{-1/2}\exp\biggl\{ -k t_c/r+(x-d)k^{1/2}/r+\biggl(-(x-d)^{2}/2\sigma^2 t_c\biggr)\biggr\},NEWLINE\]NEWLINE NEWLINE\[NEWLINEE(T_c)=(x-d)/2(2k/ q \sigma^4)^{1/2},\quad \text{Var} T_c=(x-d)/4(2k/ q \sigma^4)^{3/2}.NEWLINE\]NEWLINE The author presents the maximum-likelihood estimation of the first passage distribution.